On‐line partitioning of the sample space in the regional adaptive algorithm

Abstract

The regional adaptive (RAPT) algorithm is particularly useful in sampling from multimodal distributions. We propose an adaptive partitioning of the sample space, to be used in conjunction with the RAPT sampler and its variants. The adaptive partitioning consists in defining a hyperplane that is orthogonal to the line joining averaged coordinates in two separate regions and that goes through a point such that both averaged coordinates are equally Mahalanobis-distant from this point. This yields an adaptive process that is robust to the choice of initial partition, stabilizes rapidly and is implemented at a marginal computational cost. The ergodicity of the sampler is verified through the simultaneous uniform ergodicity and diminishing adaptation conditions. The approach is compared to the RAPT algorithm with fixed regions and to the RAPT with online recursion (RAPTOR) through various examples, including a real data application. In short, our main contribution is the development of an alternative version of RAPTOR that seems to have no obvious downside and runs 15-35% faster in the examples considered.